Find a sufficient condition that the iterated limits of a 2 variable function are interchangeable. Suppose $f$ is a function from $\mathbb R\times \mathbb R$ to $\mathbb R$.What is the condition that $\lim_{x\to a}\lim_{y\to b}f(x,y)$ and $\lim_{y\to b}\lim_{x\to a}f(x,y)$are equal?[Note that $a,b\in \mathbb R$$\cup$ {$-\infty,\infty$}]
Also it would be helpful if one suggest a quick method/rule of hand to construct such type of functions in which those limits can be interchanged.
 This may be a lot helpful in constructing counterexamples in analysis.
 A: A sufficient condition is the following:
If $\lim_{x \rightarrow a} f(x,y)$ extists for all $y$ and $\lim_{y \rightarrow b} f(x,y)$ extists for all $x$ and additionally $\lim_{y \rightarrow b} f(x,y)$ converges uniformly in a neighbourhood of $a$
(or exchanging coordinates: $\lim_{x \rightarrow a} f(x,y)$ converges uniformly in a neighbourhood of $b$).
Then $\lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x,y)$ and $\lim_{x \rightarrow a} \lim_{y \rightarrow b} f(x,y)$ exist and are equal.
You can find this in a more general setting of metric spaces in http://elib.mi.sanu.ac.rs/files/journals/tm/14/tm812.pdf, Theorem 1.
Or in Terence Tao, Analysis II, Prop 3.3.3 (3rd edition).
In the special case, where one of the limits is an integral or a series, we also have the following (one observes that a series is nothing more than a integral over the nonnegative integers with the counting measure):
Monotone convergence, Dominated convergence and the more general situation of the Vitali convergence theorem.
To point out the relation between integral and series, consider the following observation:
Let $(\mathbb{Z}_{\geq 0}, \mathcal{P}(\mathbb{Z}_{\geq 0}), \mu)$, be the measure space, where $\mu$ is the counting measure (which is $\sigma$-finite), then a function $f\colon \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}$ is the same as a sequence of real numbers and $\int_{\mathbb{Z}_{\geq 0}} f \text{d}\mu= \sum_{k=0}^{\infty}f(k)$. Hence all theorems above apply in the situation of series.
